[ ∗ note: most of the following examples are used in the future upcoming 'Relativity Science Calculator' Mac software application ]

§ Determining the mass of the earth:

§ Determining earth's velocity around the sun:

§ Determining the mass of the sun:

NASA: Giant solar eruption - April 16, 2012

The centripetal force exerted by the sun keeping earth in its near perfect circular orbit is equal but opposite to earth's centrifugal force as follows

And because of Newton's Law of Universal Gravitation, we equate the following:

Notice that if we take the ratio of sun mass to earth mass we get the following:

which means that the sun's mass is at least 300,000 times greater than earth's!!

§ Determining the mass of any other solar system planet:

Simply use the following Newton equation where the sun's mass and centripetal force will determine the planet's orbital velocity as follows

which is equivalent to

Now, in our solar system all of the planets except Mercury, Mars and the outer, dwarf - planet Pluto are almost perfectly circular. And for these non - circular elliptical orbits, radius is simply replaced by the semi-major axis of the orbital ellipse.

Finally, notice that Newton's equations validate Kepler's 3rd Law ( Harmonic Law ) - i.e., the square of a planet's orbital period is directly proportional to the cube of the planet's mean distance ( semi-major axis of the planet's elliptical orbit ) from the sun because of this simple rearrangement of the above terms:

§ The best method for determining the mass and surface acceleration of any solar system body:

Deep Impact's July 4, 2005 encounter with comet 9P/Tempel 1. When the impactor separated and flew into Tempel 1, Deep Impact spacecraft was orbiting at about 10,000 km above Tempel 1's surface. Both Deep Impact spacecraft and Tempel 1 at time of impact was approximately 0.89 AU from Earth and 1.5 AU from the Sun, the comet's perihelion elliptical distance.

Given today's Space Age technology, by far the best method for determining the mass of any solar system body such as an asteroid object, is by means of placing an artificial orbiting satellite into a near - perfect circular orbit of known radial distance to the object's center of mass and astronomically observing its period of revolution T about the object's body. Then by employing Kepler's 3rd Law ( Harmonic Law )

and knowing the artificial satellite's mass, , we get

Since, this effectively yields

And from instrument measurements onboard the artificial satellite as well as astronomical observations of the planet, it's possible to further determine the planet's radius, , and at the planet's surface any mass has weight W and acceleration g as follows

§ Determining the distance of the earth to the moon:

Apollo 17 astronauts Gene Cernan and Jack Schmitt sing while walking on the moon during the last Apollo lunar landing mission.
This was filmed in December1972.Charlie Duke - Proud 10th Man to Walk the Moon on Apollo 16, April 20, 1972,The Moon Descartes Highlands

In this calculation we basically only need to know what is the time period for the moon's rotational orbit from astronomical observation together with Euclid's geometry and apply the following Kepler's 3rd Law ( Harmonic Law ) as validated by Newton's Universal Law of Gravitational Attraction:

§ Determining mass of earth's moon:

View from the Far Side of the Moon: 1st ever video of the dark side of the Moon, Jan. 19, 2012, a joint venture of NASA's GRAIL Project, JPL and MIT.

As was mentioned above, the very best way of determining the mass of any solar system body is by placing an orbiting artificial satellite around that object and measure its period and distance to the center of the body - e.g. the moon.

Notice, again, that essentially all we need to know to determine the mass of the moon are the astronomically observed orbital moon period ( sidereal month ) and the earlier derived quantities, and earth mass, !!

(iii). The accepted derived quantities for earth and moon therefore are

Since this again involves a 2 - body system analysis, it's recommended that you first visit "The Two - Body Problem" web page .

Notice: the earth - moon barycenter is approximately 3/4 of earth's radius and resides 1/4 of the way inside earth's crust!

§ Determining the masses of near and distant disk galaxies:

Amazingly, we still use Newton's and Kepler's Laws of motion! This is accomplished first by astronomical observations of the outer most "spiral tail" of disk galaxies to determine both the radial distance to the galactic center as well as the period of rotation about the galactic center for these tails. Of course, both in terms of distances and rotational periods these quantitative elements are neither easy to obtain nor are they anything within normal human experience. In fact, everything regarding cosmic expanses is totally behind normal human experience. But the further amazing thing is that these cosmic quantities and their related cosmic galaxies are not beyond human knowledge and understanding!! This in turn leads to applying Newton's Law of Gravitational Attraction and Kepler's 3rd ( Harmonic ) Law as we previously did in the following manner:

The problem with determining the mass of our Milky Way ( The Galaxy; Latin: "Via Lactea" derived from Greek word "Kiklios Galaxios" which translates as "milky circle" ) is that we can't exactly ’see‘ our Milky Way tail since our platform for observation is Earth within our own solar system. Our solar system in turn is traversing a nearly circular orbit within the inner rim of the Orion Arm of The Galaxy at about 40 - 50% distance ( about 30,000 light - years ) away from the Galactic Center, at about 15 light - years above the plane of the Milky Way disk, with an approximate velocity of 220 - 234 kilometer per second, which is equivalent to one (1) light year in about 1,400 earth - years or one (1) AU every 8 earth days. The Sun, and hence the Earth, completes one "galactic year" in about 225 - 250 million earth - years and has made 27 round trips since its earliest formation. Nevertheless, Newton's and Kepler's Laws determine Milky Way galactic mass and with some extra luminosity observations we can further tweak the mathematics for Milky Way mass.

Notwithstanding what amounts to a "galatic year" for earth's solar system, there is now a seminal study by the astrophysicists at the Cardiff Center for Astrobiology, Cardiff, Scotland, proposing that earth's solar system transits the plane of the Milky Way galaxy approximately every 35 - 40 million years with the consequent result of sending comet collisions into the earth itself on a regular 35 - 40 million year time scale. The meaning of this is that astrobiologists as well as earth scientists can now better understand the periodicity for crater occurrences on earth's surface as well earth's recurring mass extinctions, especially for the dinosaurs some 65 million years ago. source: http://www.world-science.net/othernews/080503_galaxy

Milky Way Galaxy

Artist's concept of the Milky Way galaxy, with the "galactic bar" visible in the center. (Image by R. Hurt) source: NASA Press Release by Bob Silberg/PlanetQuest, February 14, 2006

Still, from greater astronomical observations of near and distant galaxies we discern that spiral and disk tails are traveling at doppler - shifted velocities [ note: using hydrogen's 21 cm radiation line ] greater than can be justified by the observed amount of light emanating from these galactic masses. That is, the strength of gravitational attraction between masses of bodies within any galaxy will determine spiral tail or other disk velocities. Therefore since there is such a wide disconnect or disparity in so many galaxies as between observed velocities which otherwise would tear these galaxies apart and observed [ light illuminated ] total mass of bodies providing "galactic [ gravitational ] glue", hence logically there must exist within so many galaxies hidden or dark gravitational forces which in turn can only arise from hidden or "dark matter".

Instead of WIMPS, weakly interacting massive particles, or axions, rather dark matter may be massive, according to physics professors Glenn Starkman of Case Western Reserve University and David Jacobs of University of Cape Town, in their joint paper "Macro Dark Matter".